Cannot get rid of shadow under transparent texture
The texture in your picture is not fully transparent. Since the Transparent node is set to fully white, your texture is not fully transparent in its background. Otherwise the material setup is fine.

Notice the eclipse shadow only appears behind the texture and is the same color as the surrounding shadow. This means you probably have an error in your light path setup. A simple reason could be that the transparency passes is set to low, and Cycles cuts off the light rays not allowing them to pass through the transparent material often enough.

Fix this by increasing your Transparency bounces in Properties >Render >Light Paths: